A detailed study of nuclear structure of odd-mass Pm isotopes near N = 82 shell closure

Abstract

A systematic study of the positive as well as negative parity yrast structure for the odd-mass ^145−149Pm isotopes is carried out by using the angular momentum projection technique implemented in the projected shell model (PSM). By assuming an axial symmetry in the deformed basis, we have been able to calculate the yrast states up to a maximum spin of 47/2 for both positive and negative parity, and as a result, a good agreement is obtained with available experimental data for all isotopes. The intrinsic structure of positive and negative parity yrast states has been found to be built on πg_7/2 $\otimes$ νh_11/2 and πh_11/2 $\otimes$ νh_11/2 configurations, respectively. The phenomenon of back-bending has also been discussed in the present work. Reduced electric and magnetic transitions probabilities [B(E2) and B(M1)] are also calculated using the PSM wave functions. Since experimental data on B(E2) and B(M1) are not available, these results can serve as motivation for experimentalists to look for these data. However, the theoretical values of transition strength ratio B(M1)/B(E2) have been found to be matching excellently with the experimental ones.

1 Introduction

The rare-earth nuclei with N = 88–90 are well known to have shape transition from nearly spherical or weakly deformed shapes for N < 88 to well-deformed shapes for N > 90. The nuclei with N = 88–90 lie in a transition region where shape changes from spherical to quadrupole prolate deformation. The successful application of shell model on the nuclei lying near N = 82 shell closure, to interpret the high-spin states of several nuclei in this region, shows that nuclei in this mass region are sufficiently spherical to allow a single-particle model description of their properties [1,2,3,4]. Further, the many-body nuclear correlations in this region can play an important role and lead to various shapes other than those that are spherically symmetric like reflection symmetric prolate or oblate shapes having quadrupole deformation, triaxial shapes, etc. Thus, this region has proved to be a fruitful testing ground for the study of single-particle excitations at high angular momenta. Long time ago, it has been recognized that octupole collectivity is an important component of nuclear excitations in nuclei around N = 82 region [5]. In heavier lanthanides, this collectivity gives rise to a possible octupole deformation [6,7,8,9]. In lighter isotopes, it manifests itself by octupole vibrations coupled to multi-particle configurations. A variety of such excitations have been found in the N = 85, even-Z isotones and in ¹⁴⁸Eu [8]. A similar situation is expected in the Pm isotopes as previous study of high-spin states [10] in promethium (Pm) isotopes with Z = 61 nuclei provided evidence that the yrast spectrum of Pm isotopes can be interpreted in terms of the weak coupling of unpaired d_5/2, g_7/2 and h_11/2 proton holes and protons to a Nd core. Moreover, promethium isotopes lie in the region of transitional lanthanides where evidence has been found suggesting the presence of octupole deformation [8]. Therefore, one may expect that its excitation pattern will be dominated by single-particle configurations and vibration like collective excitations. Pronounced octupole excitations, observed in the neighbouring nuclei (like ¹⁴⁹Tb [11]), suggest that octupole collectivity may play an important role in the excitation mechanism of odd-mass Pm nuclei having neutron number between 82 and 90. Further, the heavier isotopes of rare-earth region, which are mostly neutron rich, can be accessed by the fission process. The spectroscopy of heavy fission fragments provides the opportunity to investigate deformation effects as a function of neutron number for a particular isotopic chain. Pm nuclei, with a wide isotopic range, are good candidates to explore the evolution of deformation with neutron number and possible role of deformation in this region. It may be noted that there are no stable isotopes of Pm, and moreover, the isotopes beyond ¹⁵¹Pm are known only through radioactive decay studies [12,13,14,15,16,17,18,19,20]. Extensive studies of odd-A nuclei, in the past, have resulted in a good understanding of the coupling between an unpaired valence nucleon and an even–even core nucleus in cases where the core nucleus is either well deformed or spherical. But particle–core coupling behaviour is not so well understood in nuclei which lie in between these two limiting shapes. However, various studies were made in the past to understand this behaviour. Ch. Vieu et al. [21] studied high-spin states in ¹⁴⁵Pm with TESSA3 operating at NSF Daresbury, through the ¹³⁴Xe (¹⁵N, 4n) compound nucleus reaction using a solid xenon target and proposed an extended level scheme of ¹⁴⁵Pm. W. Urban et al. [22] studied excited states in ¹⁴⁷Pm through the ¹³⁶Xe (¹⁵N, 4n) and ¹⁴⁸Nd (d, 3n) compound nucleus reactions. A solid xenon target has been used to populate high-spin states in ¹⁴⁷Pm. The half-life of the isomeric level at 649.3 keV has been remeasured to be 27 ± 3 ns. An extended level scheme is proposed, which is discussed in terms of single-particle excitations and collective quadrupole and octupole vibrations coupled to them. The possible presence of octupole deformation is considered. M.A. Jonnes et al. [23] have studied the transitional lanthanide nucleus ¹⁴⁹Pm through the reaction ¹⁵⁰Nd (d, 3n). The level scheme of this nucleus was extended up to 27/2⁻, and the excited levels have been found to build over πh_11/2 state.

In our previous work [24], we have already described the structure of ^131−139Pm isotopes and all these isotopes lie below the N = 82 shell closure. Moreover, in our recent work on ^151−161Pm isotopes [25], we have successfully not only confirmed the recently reported experimental/theoretical data but also extended the already available information on the energy levels and added new information on the reduced transition probabilities. It should be noted that all Pm isotopes with A > 151 lie above the N = 92 subshell closure. Thus, the main motive to carry out the present piece of work is to study the deformation trend in Pm isotopes lying between N = 82 shell closure and N = 92 subshell closure. Moreover, the presence of Z = 64 subshell closure [26] makes ^145,147,149Pm isotopes (Z = 61) interesting candidates for investigating onset of deformation near subshell/shell closure. With this motivation, we have carried out PSM calculations on odd-mass ^145−149Pm and have analysed the calculated results in the context of physics. Various properties obtained through PSM calculations are compared with the available experimental data. The paper is arranged as follows. An outline of the PSM is given in Sect. 2. Theoretical discussion and comparison with experimental data are presented in Sect. 3. Finally, conclusions are drawn in Sec. 4.

2 An outline of theory used

In this section, we have presented briefly the formal aspects of projected shell model (PSM). For more details, one can refer to the review article on PSM [27, 28]. PSM calculations are of the shell model type, but, unlike shell model, it uses the deformed Nilsson potential. The Nilsson parameters κ and μ, used in the Nilsson potential for the construction of deformed basis, are taken from Ref. [29]. Three major shells (N = 3, 4 and 5) for both neutrons and protons are taken for the formation of configuration space. The inclusion of deformed basis has the advantage of incorporating important nuclear pairing correlations more readily. Further, with the inclusion of deformed basis, rotational symmetry is violated which is restored by the standard angular momentum projection technique [30]. Besides, the pairing correlations are added by performing BCS calculations.

In this work, we have used the following Hamiltonian:

$$ \hat{H} = \hat{H}_{0} - \frac{1}{2}\chi \mathop \sum \limits_{\mu } \hat{Q}_{\mu } Q_{\mu }^{\dag } - G_{M} \hat{P}^{\dag } \hat{P} - G_{Q} \mathop \sum \limits_{\mu } \hat{P}_{\mu }^{\dag } \hat{P}_{\mu } $$

(1)

where $\hat{H}_{0}$ is the spherical single-particle shell model Hamiltonian which contains proper spin–orbit coupling whose strengths (i.e. Nilsson parameters $\kappa$ and μ) are taken in such a manner that they reproduce the best fit of available experimental data [21,22,23, 31,32,33]. The second term in the Hamiltonian denotes the quadrupole–quadrupole two-body interaction and $\chi$ represents its strength. The value of $\chi$ is adjusted by the self-consistent relation in such a way that the input quadrupole deformation $\epsilon_{2}$ and the one resulting from the HFB procedure agrees with each other. The third and fourth terms represent monopole–pairing and quadrupole–pairing forces, respectively. The pairing strengths generally influence the level spacing within a band, and in the present calculations, their values are adjusted in such a way that known energy gap can be generated. The parameter $G_{M}$ represents the monopole–pairing strength and is defined as

$$ G_{M} = \left[ {G_{1} \mp G_{2} \frac{N - Z}{A}} \right]A^{ - 1} \left( {\text{MeV}} \right) $$

(2)

with “−” for neutrons and “+” for protons. The choice of these strengths depends on the size of the single-particle space in the calculations. In the present calculations, for both positive and negative parity, the values of $G_{1}$ and $G_{2}$ are taken as 21.20 and 12.70, respectively. The quadrupole–pairing strength G_Q is assumed to be proportional to G_M, with the proportionality constant being fixed as 0.16 for positive parity and 0.18 for negative parity in the present calculations. The parameters of the basis deformation $\epsilon_{2}$ and $\epsilon_{4}$ used in our calculations for each nucleus are given in Table 1. Different configurations (one and three quasi-particle states) are built within the chosen basis.

Table 1 Deformation parameters used as input in the present calculations of ^145−149Pm nuclei

3 Results and discussion

In the present calculations, the valance space is truncated by the inclusion of the states within an energy window of 3.5 MeV, both for protons and for neutrons, around the Fermi surface. This determines the size of the basis space. In the next step, these basis states are projected to good angular momentum states and the projected basis is then used to diagonalize the shell model Hamiltonian which, in turn, gives rise to the yrast energy spectra. Further, the resulting wave functions are subsequently used to calculate various nuclear structure properties including reduced transition probabilities. These results along with some other nuclear structure properties obtained through PSM calculations and their comparison with the experimentally available data are presented in the following subsections.

3.1 Back-bending in moment of inertia

The phenomenon of back-bending in moment of inertia is an important feature of rotational spectra of deformed nuclei. Since yrast band is formed by states of both 1-qp (which generally forms the lower spin states) and 3-qp bands (which form the major components of the higher spins), the phenomenon of back-bending carries important information on the interplay and alignment between the 1-qp band and multi-qp bands. In Fig. 1, we have made comparison of the PSM results with experimental data for positive as well as negative parities states for odd-mass^145−149Pm isotopes. We plot twice the kinetic moment of inertia $2{\Im }^{\left( 1 \right)}$ as a function of square of rotational frequency (ћ² $\omega$ ²). These quantities are defined as [34].

Fig. 1

PSM results for twice the kinetic moment of inertia [2ℑ⁽¹⁾(ћ²MeV⁻¹)] plotted against the angular frequency squared (ħ²ω²) for positive parity a ¹⁴⁵Pm, b ¹⁴⁷Pm and c ¹⁴⁹Pm and negative parity d ¹⁴⁵Pm, e ¹⁴⁷Pm and f ¹⁴⁹Pm. The experimental data [21,22,23, 31,32,33] are also shown for comparison

$$ 2{\Im }^{\left( 1 \right)} = \frac{{\left( {2I - 1} \right)}}{\omega } , $$

(3)

$$ \omega = \frac{{E_{\gamma } }}{{\sqrt {\left( {I + 1} \right)\left( {I + 2} \right) - K^{2} }^{ } {-}\sqrt {\left( {I - 1} \right)I - K^{2} }^{ } }} $$

(4)

where $E_{\gamma } = E\left( I \right) - E\left( {I - 2} \right) .$

3.1.1 Positive parity

For ¹⁴⁵Pm (Fig. 1a), the experimental and theoretical back-bending occur at the spin value of 15/2⁺ and 11/2⁺ corresponding to almost the same values of rotational frequencies of 0.1611 ћ² $\omega$ ² and 0.1756 ћ² $\omega$ ², respectively. On the other hand, for ¹⁴⁷Pm (Fig. 1b), the experimental back-bending in moment of inertia is reproduced exactly at the same value of spin 15/2⁺ by PSM results and this spin value also corresponds to almost the same rotational frequencies of 0.1052 ћ² $\omega$ ²and 0.1044 ћ² $\omega$ ², respectively. By examining the moment of inertia pattern for ¹⁴⁹Pm in Fig. 1c, back-bending is observed to occur experimentally at the spin I = 15/2⁺corresponding to the rotational frequency of 0.0885 ћ² $\omega$ ², whereas PSM calculations predict the occurrence of back-bending corresponding to rotational frequency of 0.1470 ћ² $\omega$ ² at the spin I = 19/2⁺.

3.1.2 Negative parity

From Fig. 1d, it is found that, for negative parity ¹⁴⁵Pm, the experimental back-bending is observed at the spin value of 19/2⁻ corresponding to rotational frequency of 0.0578 ћ² $\omega$ ², whereas the PSM calculations predict the back-bending at spin 21/2⁻ corresponding to rotational frequency of 0.1059 ћ² $\omega$ ², respectively. For ¹⁴⁷Pm, in Fig. 1e, experimentally observed back-bending is found to occur at the spin 23/2⁻ corresponding to the rotational frequency of 0.0971 ћ² $\omega$ ² whereas PSM calculations also predict a back-bending but a little delayed at the spin 31/2⁻ corresponding to rotational frequency 0.1558 ћ² $\omega$ ². Further, for ¹⁴⁹Pm (Fig. 1 f), the PSM calculations predict the back-bending at the spin 27/2⁻ at rotational frequency 0.1033 ћ² $\omega$ ², while, due to less experimental data available in this nucleus, no experimental back-bending could be found to occur in this nucleus.

It is quite noteworthy from Fig. 1 that the comparison between theory and experiment is overall satisfactory. The observed variation in ${\Im }^{\left( 1 \right)}$ in odd-mass ^145−149Pm isotopes is an important indication for structural changes in wave functions as it is well known that nucleons in high-j orbits respond sensitively to the rotation and can easily align their spin direction along with the system rotation, and this observation is successfully reproduced by PSM calculations. This change in the moment of inertia $\left( { {\Im }^{\left( 1 \right)} } \right)$ with rotational frequency may arise due to the interaction between various 1-qp and 3-qp bands which will be discussed thoroughly in the next subsection. Furthermore, the positive parity experimental data show several back-bending phenomena in ^145−147Pm, whereas the theoretical results are not enough to reproduce these phenomena, as shown in Fig. 1. This could be because of the reason that the valence space of the present calculations is not enough to describe the high-spin states from 33/2⁺.

3.2 Analysis of band diagrams

The results of back-bending phenomenon in odd-mass ^145−149Pm isotopes indicate the presence of rotational alignment of quasiparticles in particular orbitals of these nuclei. For odd-mass nuclei, it is associated with crossings between bands with 1-qp and 3-qp configurations. In the present work, 1-qp configurations arise from the deformed proton g_7/2 orbit for positive parity and h_11/2 orbit for negative parity, while 3-qp ones consist of these proton 1-qp configurations plus a pair of h_11/2 neutrons. Thus, the plot of energies of various angular-momentum-projected multi-quasi-particle bands as a function of spin, generally known as band diagrams, helps us to understand the back-bending effects which are found to be present (discussed above as per Fig. 1) in the odd-mass ^145−149Pm isotopes.

In Fig. 2, band energies of each angular-momentum-projected configuration are plotted as functions of spin for both positive and negative parity bands. The energy of a band κ is defined by

Fig. 2

Band diagrams for positive parity a ¹⁴⁵Pm, b ¹⁴⁷Pm and c ¹⁴⁹Pm and negative parity d ¹⁴⁵Pm, e ¹⁴⁷Pm and f ¹⁴⁹Pm

$$ E_{\kappa } \left( I \right) = \frac{{\left\langle {\phi _{\kappa } \left| {\hat{H}\hat{P}_{{KK}}^{I} } \right|\phi _{\kappa } } \right\rangle }}{{\left\langle {\phi _{\kappa } \left| {\hat{P}_{{KK}}^{I} } \right|\phi _{\kappa } } \right\rangle }} $$

(5)

Here, $\hat{P}_{KK}^{I}$ is the angular momentum projection operator.

3.2.1 Positive parity

For ¹⁴⁵Pm, it is clear from Fig. 2a that at low spins yrast states up to spin 11/2⁺ are formed by 1-qp band 1πg_7/2 [5/2], K = 5/2. However, as spin increases from 11/2⁺ to 13/2⁺, this 1-qp band is approached by several 3-qp bands and band crossing occurs at spin 13/2⁺. These 3-qp bands are identified as 1πg_7/2 [5/2] $+$ 2νh_11/2 [1/2, − 3/2], K = 3/2, 1πg_7/2 [− 5/2] $+$ 2νh_11/2 [− 1/2, − 3/2], K = − 7/2, 1πg_7/2 [5/2] $+$ 2νh_11/2 [1/2, 1/2], K = 7/2 and 1πg_7/2 [− 3/2] $+$ 2νh_11/2 [1/2, 1/2], K = − 1/2. These superimposed 3-qp bands continue to form yrast spectra up to spin 17/2⁺. After that, i.e. from spin 19/2⁺ onwards, the 3-qp band 1πg_7/2 [5/2] $+$ 2νh_11/2 [1/2, 1/2], K = 7/2 remains lowest in energy and leads to the formation of yrast states.

It can be seen from Fig. 2b that the yrast states for ¹⁴⁷Pm up to spin 19/2⁺ are found to build on the 1-qp band 1πg_7/2 [5/2], K = 5/2. At the next spin, i.e. 21/2⁺, a strong interaction between the 1-qp band and a 3-qp band is seen which results in band crossing. This interacting 3-qp band forms the yrast spectra up to the spin 31/2⁺ and is identified as 1πg_7/2 [5/2] $+$ 2νh_11/2 [1/2, 1/2], K = 7/2. Afterwards, another 3-qp band identified as 1πg_7/2 [− 3/2] $+$ 2νh_11/2 [− 11/2, 1/2], K = − 13/2 appears as main contributor towards the formation of yrast spectra for the rest of the spin values. Moreover, two more 3-qp bands having configurations 1πg_7/2 [− 3/2] $+$ 2νh_11/2 [1/2, − 3/2], K = − 5/2 and 1πg_7/2 [5/2] $+$ 2νh_11/2 [1/2, − 3/2], K = 3/2 are also found lower in energy and shown in Fig. 2b.

Talking about ¹⁴⁹Pm (Fig. 2c), for the spin range I = 7/2⁺ to 21/2⁺, the yrast band has the main component of the two 1-qp bands 1πg_7/2 [5/2], K = 5/2 and 1πg_7/2 [− 3/2], K = − 3/2. This configuration is strongly disturbed by a 3-qp band 1πg_7/2 [5/2] $+$ 2νh_11/2[1/2, 1/2], K = 7/2, and the band crossing occurs at spin 23/2⁺. As the spin increases beyond 23/2⁺, few more 3-qp’s with configurations 1πg_7/2 [− 3/2] $+$ 2νh_11/2 [1/2, − 3/2], K = − 5/2, 1πg_7/2 [− 3/2] $+$ 2νh_11/2 [1/2, 1/2], K = − 1/2 and 1πg_7/2 [− 9/2] $+$ 2νh_11/2 [5/2, − 3/2], K = − 7/2 bands dive down in energy and contribute towards the formation of yrast spectra up to the last calculated spin.

3.2.2 Negative parity

It is clear from the band diagram for ¹⁴⁵Pm, shown in Fig. 2d, that up to the spin 23/2⁻, the yrast band is composed of 1-qp band, 1πh_11/2[− 3/2], K = − 3/2, which is then crossed by a 3-qp bands with configuration: 1πh_11/2[− 11/2] $+$ 2υh_11/2[− 3/2, 1/2], K = − 13/2. Thus, band crossing occurs at spin 23/2⁻. This 3-qp band along with two more 3-qp bands: 1πh_11/2[1/2] $+$ 2υh_11/2[− 11/2, 1/2], K = − 9/2 and 1πh_11/2[1/2] $+$ 2υh_11/2[− 11/2, − 3/2], K = − 13/2, are found to be lower in energy contributing to the yrast energy levels up to the last calculated spin.

In the case of ¹⁴⁷Pm (Fig. 2e), the yrast band up to a spin of 33/2⁻ is built by1-qp 1πh_11/2[− 3/2], K = − 3/2. As we reach the spin 35/2⁻, various 3-qp bands approach the above-mentioned 1-qp band and the interaction becomes maximum at 35/2⁻, where band crossing occurs, and the 3-qp band 1πh_11/2[− 3/2] $+$ 2υh_11/2[− 11/2, 1/2], K = − 13/2 crosses the 1-qp band and forms the rest of the yrast spectra along with two more 3-qp bands: 1πh_11/2[1/2] $+$ 2υh_11/2[− 11/2, − 3/2], K = − 13/2 and 1πh_11/2[1/2] $+$ 2υh_11/2[− 11/2, 1/2], K = − 9/2.

Finally, for ¹⁴⁹Pm (see Fig. 2f), up to the spin of 31/2⁻, the yrast spectra arise because of superimposition of two 1-qp bands, 1πh_11/2[1/2], K = 1/2 and 1πh_11/2[− 3/2], K = − 3/2. At spin 31/2⁻, band crossing occurs and a 3-qp band having configuration 1πh_11/2[1/2] $+$ 2υh_11/2[1/2, 1/2], K = 3/2 becomes lower in energy and contributes towards the formation of yrast along with two more 3-qp bands having configurations 1πh_11/2[1/2] $+$ 2υh_11/2[1/2, − 3/2], K = − 1/2 and 1πh_11/2[− 3/2]$+$ 2υh_11/2[1/2, 1/2], K = − 1/2 up to the last calculated spin.

Further, from the above discussion on band diagrams, we can say that the observed back-bending in ^145,147,149Pm isotopes may be described as being happening due to the crossing of the ground band, i.e. 1-qp band consisting of single proton (1πg_7/2 for positive parity and 1πh_11/2 for negative parity) with another 3-qp rotational band consisting of one proton and two neutrons (1πg_7/2 + 2υh_11/2 for positive parity and 1π h_11/2 + 2υh_11/2 for negative parity) having a larger moment of inertia.

3.3 Yrast spectra

In the present work, PSM calculations have been carried out by assuming an axial symmetry in the deformed basis where each band (1-qp or 3-qp) can be assigned with a K quantum number. We have already discussed the various 1-qp and 3-qp bands and the interaction between them which leads towards the formation of yrast spectra both for the positive and for the negative parity. The yrast spectra are obtained by diagonalization of the Hamiltonian in the deformed basis constructed using Nilsson potential, which, in turn, corresponds to the configuration mixing or band mixing in shell model. Calculated yrast energy levels for both positive and negative parity of odd-mass ^145−149Pm isotopes are plotted against the spin in Figs. 3 and 4, respectively. Further, to check the degree of accuracy, a comparison of these calculated results is made with the experimentally available data [23,24,25, 33,34,35] in the same figures. These comparisons show the efficacy of the applied PSM technique as the experimental data are very well reproduced.

Fig. 3

Comparison of calculated and experimentally available yrast spectra for positive parity a ¹⁴⁵Pm, b ¹⁴⁷Pm and c ¹⁴⁹Pm. The experimental data [21,22,23, 31,32,33] are also shown for comparison

Fig. 4

Comparison of calculated and experimentally available yrast spectra for negative parity a ¹⁴⁵Pm, b ¹⁴⁷Pm and c ¹⁴⁹Pm. The experimental data [21,22,23, 31,32,33] are also shown for comparison

Theoretically, we have been able to obtain the data on positive and negative parity yrast states in these nuclei up to a maximum spin of 47/2, whereas experimental data on the yrast states for the positive parity are available up to a maximum spin value 39/2⁺, 41/2⁺ and 23/2⁺ for ¹⁴⁵Pm, ¹⁴⁷Pm and ¹⁴⁹Pm, respectively, and 27/2⁻ for negative parity of all the three isotopes ¹⁴⁵Pm, ¹⁴⁷Pm and ¹⁴⁹Pm. Experimentally, the band heads for positive parity states of ¹⁴⁵Pm, ¹⁴⁷Pm and ¹⁴⁹Pm have been observed as 5/2⁺, 7/2⁺ and 7/2⁺, respectively, whereas for negative parity states, the band head spin of all these ^145−149Pm has been measured as 11/2⁻. In our calculations, we are successful to reproduce the band heads for both positive and negative parities yrast bands of odd-mass ^145−149Pm isotopes. Moreover, by analysing the band diagrams, we can say that the proposed band structures in the present work are built on 1-qp bands at low spins, whereas at higher spins, the yrast states are built on 3-qp bands.

3.4 Analysis of the wave functions

For better evaluation of nuclear structure, one requires a detailed analysis of the wave functions. In the present PSM calculations, the eigenvalues of energy along with the amplitude of wave functions have been obtained by diagonalization of the total Hamiltonian. In Fig. 5, we have presented the average of the amplitudes of the wave functions corresponding to various 1-qp and 3-qp bands responsible for the formation of positive as well as negative parity yrast spectra of odd-mass ^145−149Pm isotopes. The interactions between wave functions of 1-qp bands and 3-qp bands, shown in these plots, clearly depict the origin of certain characteristics phenomena in these nuclei, like back-bending in moment of inertia, band crossing, configuration mixing, etc. (explained earlier in the previous subsections). It should be noted that average amplitudes of the wave functions corresponding to 1-qp bands cross the average amplitude of wave functions corresponding to 3-qp bands at the same values of spins at which band crossing takes place. Further, the trends of probability amplitudes show that yrast bands for ^145−149Pm isotopes have well-defined structures built on 1-qp configurations at lower spins, but at higher spins, the yrast states are dominated by 3-qp bands. Further, it is quite clear that the experimental negative parity yrast states are quite well reproduced for the entire spin range by the present PSM calculations, but, for positive parity states, there is a little variation in the data in the high-spin regime. Thus, the present calculated PSM wave functions are well enough to reproduce and explain the various observed nuclear structure phenomena at lower spins in odd-mass ^145−149Pm nuclei. With this motivation, we have used the wave functions obtained through PSM calculations to calculate the reduced transition probabilities which are discussed in the next section.

Fig. 5

Probability amplitude of various projected K-configurations in the wave functions of the positive parity yrast bands for a ¹⁴⁵Pm, b ¹⁴⁷Pm and c ¹⁴⁹Pm and the negative parity yrast bands for d ¹⁴⁵Pm, e ¹⁴⁷Pm and f ¹⁴⁹Pm

3.5 Reduced transition probabilities

The reduced electric quadrupole transition probability B(E2) (from an initial state (I_i = I) to a final state (I_f = I − 2)) is defined by

$$ B\left( {E2, I \to I - 2} \right) = \frac{{e^{2} }}{2I + 1}\left| {\varPsi^{I - 2} \left| {\left| {\hat{Q}_{2} } \right|} \right|\varPsi^{I} } \right|^{2} $$

(6)

where the operator $\hat{Q}_{2}$ is related to the quadrupole operators.

In the present calculations, we have taken the standard values of effective charges, i.e. 1.5e for protons and 0.5e for neutrons. The effective charges are fixed for all nuclei studied in this work without any individual adjustment. Generally, these values measure the extent up to which quadrupole distortions are present in the structure of a nucleus. The PSM results on the reduced electric transition probabilities (B(E2)) for positive as well as negative parity yrast band in ^145−149Pm isotopes are given in Tables 2 and 3, respectively. It should be noted that experimental data on B(E2) values for positive as well as negative parity yrast states on these nuclei are not available, so these results could serve as motivation for experimentalists.

Table 2 B(E2) values for positive parity ^145,147,149Pm isotopes

Table 3 B(E2) values for negative parity ^145,147,149Pm isotopes

Further, using PSM wave function, one can also calculate reduced magnetic transition probability B(M1). The reduced magnetic dipole transition probability B(M1) from initial state (I_i = I) to final state (I_f = I − 1) is given by

$$ B\left( {M1, I \to I - 1} \right) = \frac{{e^{2} }}{2I + 1}\left| {\varPsi^{I - 1} \left| {\left| {\hat{M}_{l} } \right|} \right|\varPsi^{I} } \right|^{2} $$

(7)

where the magnetic dipole operator is defined as

$$ {\hat{\rm M}}_{{_{1} }}^{\tau } = g_{l}^{\tau } \hat{j}^{\tau } + \left( {g_{s}^{\tau } - g_{l}^{\tau } } \right) \, \hat{s}^{\tau }. $$

Here, τ is either π or ν, and g_l and g_s are the orbital and the spin gyromagnetic factors, respectively. In the present work, we use the free values for g_l, whereas for g_s, the free values were damped by a factor of 0.85. The standard free values of g_l and g_s for protons and neutrons are

$$ g_{l}^{\pi } = 1,\,g_{l}^{\nu } = 0,\,g_{s}^{\pi } = 5.586 \times 0.85,\,g_{s}^{\nu } = - 3.826 \times 0.85. $$

The calculated values of magnetic transition probabilities B(M1) for positive parity yrast band of the nuclei under study are given in Table 4. The usual staggering pattern of B(M1) values can be seen from these data. Again, no experimental data on B(M1) values are available for yrast bands on these nuclei.

Table 4 B(M1) values for positive parity ^145,147,149Pm isotopes

In order to check the accuracy of the calculated B(E2) and B(M1) values, we decided to compare the theoretically calculated electromagnetic transition strength ratio (B(M1)/B(E2) with the values of transition strength ratios which are extracted from the experimentally measured intensities [33, 34] using the relation given in Ref. [35]. Figure 6 shows the comparison of experimental and calculated values of B(M1)/B(E2) ratio for the positive parity yrast band of odd-mass ^145−149Pm isotopes. The theoretical values of B(M1)/B(E2) ratio appear to agree with the measured values which authenticate the efficacy of the PSM technique and validate the calculated B(E2) and B(M1) values for the yrast band of the nuclei under study. Moreover, no experimental data are available on B(M1) and B(E2) for the yrast states, so the calculated values of these parameters in the present work will prove useful for various theoretical as well as experimental research groups.

Fig. 6

taken from Refs. [31,32,33]

Comparison of calculated B(M1)/B(E2) transition strengths ratios with the experimental values obtained from the measured intensity of transitions for positive parity bands in a ¹⁴⁵Pm, b ¹⁴⁷Pm and c ¹⁴⁹Pm. The experimental data are

4 Summary and perspectives

PSM calculations, in the present work, have been able to reproduce quite successfully the positive as well as negative parity yrast states in odd-mass Pm nuclei with neutron number 82 < N < 90. The intrinsic structure of positive and negative parity yrast states has been found to be built on πg_7/2 $\otimes$ νh_11/2 and πh_11/2 $\otimes$ νh_11/2configuration, respectively. The reduced transition probabilities, i.e. B(M1) and B(E2), have also been calculated for these nuclei. However, experimental data on B(M1) and B(E2) values for these nuclei are not available, yet we are able to confirm their accuracy by comparing their ratio (B(M1)/B(E2)) with the corresponding values obtained from experimentally measured intensities. Moreover, the calculated values of reduced transition probabilities, in the present work, would strengthen the understanding of the intricate structure of these nuclei at high spins and will serve as a motivation for the experimentalists to work for extraction of these values. The origin of back-bending phenomenon in ^145−149Pm isotopes is very well explained through band diagrams as well as by PSM wave functions